My question refers to an exercise in "Topology of Real Algebraic Sets" by Akbulut and King. This exercise is about one of the possible definition of blow-up in a differential framework, more precisely:
Let $M$ be a real smooth manifold and $L\subset M$ a smooth proper submanifold of $M$. Define the ideal:
$\mathscr{I}_M^\infty(L)=\{f\in\mathcal{C}^\infty(M):f(x)=0\,\,\text{for every }x\in L\}$.
Then, $\mathscr{I}_M^\infty(L)$ is finitely generated.
I am not so used to the properties of $\mathcal{C}^\infty(M)$ as a ring and I really do not understand where to use the hypothesis about the properness of the injection from $L$ to $M$. I am grateful to anyone will give me some hint, solution or reference about this exercise.